The theory of determinants in the historical order of development, by Sir Thomas Muir.

ALTERNANTS (CAYLEY, 1846)

163

where P stands for the difference-product of al, a2,..., am, or
for what Sylvester afterwards denoted by s1(a1, a2,.., am):
and the problem is professedly to express fm(x) " par les coefficients de f(x)," but in reality to express it as a series arranged
according to descending powers of x.
This is accomplished by partitioning P/(x- a1)(x-a) )..
(x-am) into an aggregate of fractions having x-al, x- a2,...
for denominators,- namely,

m-1 _ y, a.2,.., am)
( - -(x-a1)(x-a2)..*. (x-an

2 - a2 a,.., a.)
) a-,a3 a
2 -(a,, a,,,...a, a.
-a2 * * *

so that the coefficient of x-"' is seen to be
r-l ~[a ~.
a, t (a2, a3,. am) - a2 (al, a3, * am)
and therefore to be
m-2 r —1
1 a(. a a1
1 a2 am-2 r-l
m- 1 -l 2 a.... am 2 2 -2
l am 6a2 am-2 ar-1
i a^ a^... ^

+- ~~~

where a1, a2,..., am are the first m a's chosen from
a1, 2.,.. *
Multiplying both sides by P and performing the requisite
summation we find that the coefficient of x-r in fm(x) - f(x) is

So S1
Sm-1 Sm

m-2 8r-1... m 1 8r,

or
I

V,-1 say,.. 82m-3 8r+m-2

where Sq is the sum of the qth powers of all the a's; in other
words, that
fm(x) f(X) =- X-m V-1 + — 1 V, +.....
* It may be noted in this connection that
C2(aL, a2,...., an)0.'(Ck) = (-)nlIC2(, a2,..., a l-, a+i,..., a,)
if ((x) = (x-al)(x-a2)... (x-a ).